A class of attractions/repulsion functions for stable swarm aggregations

In this brief article, we consider an M-member 'individual-based' continuous time swarm model in an n-dimensional space and extend the results in Gazi and Passino (2003) by specifying a general class of attraction/repulsion functions that can be used to achieve swarm aggregation. These functions are odd functions that have terms for attraction and repulsion acting in opposite directions in compliance with real biological swarms. We present stability analysis for several cases of the functions considered to characterize swarm cohesiveness, size and ultimate motions while in a cohesive group. Moreover, we show how the model can be extended for achieving formation control. Furthermore, we discuss how the attraction repulsion functions can be modified to incorporate the finite body size of the swarm members. Numerical simulations are also presented for illustration purposes.